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Mid-Chapter Quiz: Lessons 7-1 through 7- 4 Graph each function. State the domain and range. 1. f (x) = 3(4) x SOLUTION: Make a table of values. Then plot the points and sketch the graph. Domain = {all real numbers}; Range = {f (x) | f (x) > 0} x 2. f (x) = −(2) + 5 SOLUTION: Make a table of values. Then plot the points and sketch the graph. Domain = {all real numbers}; Range = {f (x) | f (x) > 0} Domain = {all real numbers}; Range = {f (x) | f (x) < 5} x 2. f (x) = −(2) + 5 SOLUTION: Make a table of values. Then plot the points and sketch the graph. 3. f (x) = −0.5(3) x+2 +4 SOLUTION: Make a table of values. Then plot the points and sketch the graph. eSolutions Manual - Powered by Cognero Page 1 Domain = {all realLessons numbers}; = {f7(x)4| f (x) < Mid-Chapter Quiz: 7-1Range through 5} 3. f (x) = −0.5(3) x+2 +4 SOLUTION: Make a table of values. Then plot the points and sketch the graph. Domain = {all real numbers}; Range = {f (x) | f (x) < 4} 4. SOLUTION: Make a table of values. Then plot the points and sketch the graph. Domain = {all real numbers}; Range = {f (x) | f (x) < 4} Domain = {all real numbers}; Range = {f (x) | f (x) < 8} 4. SOLUTION: Make a table of values. Then plot the points and sketch the graph. 5. SCIENCE You are studying a bacteria population. The population originally started with 6000 bacteria cells. After 2 hours, there were 28,000 bacteria cells. a. Write an exponential function that could be used to model the number of bacteria after x hours if the number of bacteria changes at the same rate. b. How many bacteria cells can be expected after 4 hours? SOLUTION: a. Substitute 28000 for y 6000 for a and 2 for t into an eSolutions Manual - Powered by Cognero exponential function to determine the value of b. The function that models the number of bacteriaPage 2 after x hours is f (x) = 6000(2.16025)x. b. function . Domain = {all realLessons numbers}; = {f7(x)4| f (x) < Mid-Chapter Quiz: 7-1Range through 8} 5. SCIENCE You are studying a bacteria population. The population originally started with 6000 bacteria cells. After 2 hours, there were 28,000 bacteria cells. a. Write an exponential function that could be used to model the number of bacteria after x hours if the number of bacteria changes at the same rate. b. How many bacteria cells can be expected after 4 hours? SOLUTION: a. Substitute 28000 for y 6000 for a and 2 for t into an exponential function to determine the value of b. 6. MULTIPLE CHOICE Which exponential function has a graph that passes through the points at (0, 125) and (3, 1000)? x A f (x) = 125(3) B f (x) = 1000(3)x x C f (x) = 125(1000) D f (x) = 125(2)x SOLUTION: x Find f (0) and f (3) for the function f (x) = 125(2) . The function that models the number of bacteria after x hours is f (x) = 6000(2.16025)x. b. Substitute 4 for x in the function . D is the correct choice. 7. POPULATION In 1995, a certain city had a population of 45,000. It increased to 68,000 by 2007. a. What is an exponential function that could be used to model the population of this city x years after 1995? b. Use your model to estimate the population in 2020. SOLUTION: a. Substitute 68000 for y 45000 for a and 12 for t into 6. MULTIPLE CHOICE Which exponential function has a graph that passes through the points at (0, 125) and (3, 1000)? x A f (x) = 125(3) B f (x) = 1000(3)x an exponential function to determine the value of b. x C f (x) = 125(1000) D f (x) = 125(2)x SOLUTION: x Find f (0) and f (3) for the function f (x) = 125(2) . The function that models the population of the city x years after 1995 is . b. Substitute 25 for x in the function . Manual - Powered by Cognero eSolutions D is the correct choice. 7. POPULATION In 1995, a certain city had a Page 3 8. MULTIPLE CHOICE Find the value of x for log3 2 (x + 2x) = log (x + 2). function . Mid-Chapter Quiz: Lessons 7-1 through 7- 4 D is the correct choice. 7. POPULATION In 1995, a certain city had a population of 45,000. It increased to 68,000 by 2007. a. What is an exponential function that could be used to model the population of this city x years after 1995? b. Use your model to estimate the population in 2020. SOLUTION: a. Substitute 68000 for y 45000 for a and 12 for t into 8. MULTIPLE CHOICE Find the value of x for log3 2 (x + 2x) = log3 (x + 2). F x = −2, 1 G x = −2 H x =1 J no solution SOLUTION: an exponential function to determine the value of b. The function that models the population of the city x years after 1995 is . b. Substitute 25 for x in the function . Substitute each value into the original equation. Thus, x = 1. H is the correct choice. 8. MULTIPLE CHOICE Find the value of x for log3 2 (x + 2x) = log3 (x + 2). F x = −2, 1 G x = −2 H x =1 J no solution SOLUTION: Graph each function. 9. f (x) = 3 log2 (x − 1) SOLUTION: The function is a transformation of the graph of . a = 3: The graph is expanded vertically. h = 1: The graph is translated 1 unit to the right. k = 0: There is no vertical shift. Substitute each value into the original equation. eSolutions Manual - Powered by Cognero Page 4 10. f (x) = −4 log3 (x − 2) + 5 Thus, x = 1. Mid-Chapter Quiz: Lessons 7-1 through 7- 4 H is the correct choice. Graph each function. 9. f (x) = 3 log2 (x − 1) SOLUTION: The function is a transformation of the graph of . 11. MULTIPLE CHOICE Which graph below is the graph of the function f (x) = log3 (x + 5) + 3? a = 3: The graph is expanded vertically. h = 1: The graph is translated 1 unit to the right. k = 0: There is no vertical shift. SOLUTION: The function is a transformation of the graph of . 10. f (x) = −4 log3 (x − 2) + 5 SOLUTION: The function is a transformation of the graph of . h = –5: The graph is translated 5 units to the left. k = 3: The graph is translated 3 units up. So, A is the correct choice. Evaluate each expression. 12. log4 32 SOLUTION: a = –4: The graph is reflected across the x-axis. h = 2: The graph is translated 2 units to the right. k = 5: The graph is translated 5 units up. 12 13. log5 5 SOLUTION: 14. log16 4 11. MULTIPLE CHOICE Which graph below is the graph of the function f (x) = log3 (x + 5) + 3? SOLUTION: eSolutions Manual - Powered by Cognero Page 5 12 13. log5 5 SOLUTION: Mid-Chapter Quiz: Lessons 7-1 through 7- 4 2x + 3 14. log16 4 19. 16 SOLUTION: SOLUTION: 15. Write log9 729 = 3 in exponential form. < 64 20. SOLUTION: SOLUTION: Solve each equation or inequality. Check your solution. x 2 16. 3 = 27 SOLUTION: 21. SOLUTION: 3x − 1 17. 4 x = 16 SOLUTION: 22. log7 (−x + 3) = log7 (6x + 5) SOLUTION: 18. SOLUTION: Substitute x value into the original equation. 2x + 3 19. 16 < 64 SOLUTION: eSolutions Manual - Powered by Cognero Page 6 Thus, . Mid-Chapter Quiz: Lessons 7-1 through 7- 4 22. log7 (−x + 3) = log7 (6x + 5) Thus, solution set is . 24. log8 (3x + 7) = log8 (2x − 5) SOLUTION: SOLUTION: Substitute x value into the original equation. Substitute x value into the original equation. log8 (–29) is undefined, so no solution. Thus, . 23. log2 x < −3 SOLUTION: Thus, solution set is . 24. log8 (3x + 7) = log8 (2x − 5) SOLUTION: Substitute x value into the original equation. log8 (–29) is undefined, so no solution. eSolutions Manual - Powered by Cognero Page 7